POLARIZED LIGHT FROM DIFFERENT SOURCES. 247 whence we find lit (c*) = -J- {1 + sin 2/3x sin 2/3 + cos 2at cos 2a cos 2/3x cos 2/3 -f sin 2at sin 2a cos 2/3, cos 2/3} m (c2) . . .(15). There will be no occasion to write down the value of in (ca2), since ci may be taken to refer to either component. Let us pass now to the consideration of a group consisting of any number of independent polarized streams. Let 2m (c2) = A , 2 sin 2/3m (c2) = J5, 2 cos 2a cos 2/3m (c2) = (7, g Ssin2acos2/3W(c2) = D, j( '' and let cx now refer to one of the components of the whole group ; then 2m (c*) = A + B sin 2/3t + G cos 2ax cos 2ft + J5 sin 2ax cos 2/3, . . .(17). It follows that if there are two groups of independent polarized streams which are such as to give the same values to each of the four quantities A, B, C, D defined by (16), if the groups be resolved in any manner whatsoever, which is the same for both, into two oppositely polarized streams, the intensities of the components of the one group will be respectively equal to the intensities of the components of the other group. Conversely, if two groups of oppositely polarized streams are such that when they are resolved in any manner, the same for both, into two oppositely polarized streams, the intensities of the components of the one group are respectively equal to the intensities of the components of the other group, the quantities A, B, C, D must be the same for the two groups. For, if we take accented letters to refer to the second group, the second member of equation (17), and the expression thence derived by accenting A, B, 0, D, must be equal independently of al arid ft, which requires that A', Bf, G', D' be respectively equal to A, B, G, D. DEFINITION. Two such groups will be said to be equivalent. 10. The theoretical definition of equivalence which has just been given agrees completely with the experimental tests of equivalence. One of the most ready as well as delicate modes of detecting minute traces of polarization, and at the same time determining qualitatively the nature of the polarization, consists in viewing the light to be examined through a plate of calcareous